Variational for Fermions

Vijay B. Shenoy

[email protected]

Indian Institute of Science, Bangalore

December 26, 2009

1/91 Acknowledgements

Sandeep Pathak

Department of Science and Technology

2/91 Plan of the Lectures on Variational Monte Carlo Method

Motivation, Overview and Key Ideas Details of VMC Application to Superconductivity

3/91 The Strongly Correlated Panorama...

Fig. 1. Phase diagrams ofrepresentativemateri- als of the strongly cor- related electron family (notations are standard and details can be found in the original refer- ences). (A) Temperature versus hole density phase diagram of bilayer man- ganites (74), including several types of antiferro- magnetic (AF) phases, a ferromagnetic (FM) phase, and even a glob- ally disordered region at x 0 0.75. (B) Generic phase diagram for HTSC. SG stands for spin glass. (C) Phase diagram of single layered ruthenates (75, 76), evolving from a superconducting (SC) state at x 0 2 to an AF insulator at x 0 0(x controls the bandwidth rather than the carrier density). Ruthenates are believed to be clean metals at least at large x, thus providing a family of oxides where competition and com- plexity can be studied with less quenched dis- order than in Mn ox- ides. (D) Phase diagram of Co oxides (77), with SC, charge-ordered (CO), and magnetic regimes. (E) Phase diagram of the organic k-(BEDT-

TTF)2Cu[N(CN)2]Cl salt (57). The hatched re- gion denotes the co- existence of metal and insulator phases. (F) Schematic phase dia- gram of the Ce-based heavy fermion materials (51).

(Dagotto, Science, (2005))

4/91 ...and now, Cold Atom Quantum Simulators

(Bloch et al., RMP, (2009)) Hamiltonian...for here, or to go?

5/91 The Strategy

Identify “key degrees of freedom” of the system of interest Identify the key interactions Write a Hamiltonian (eg. tJ-model)...reduce the problem to a model “Solve” the Hamiltonian .... Nature(Science)/Nature Physics/PRL/PRB...

6/91 “Solvers”

Mean Field Theories (MFT) Slave Particle Theories (and associated MFTs) Dynamical Mean Field Theories (DMFT) Exact Diagonalization Methods (eg. Determinental) Variational Monte Carlo (VMC) ... AdS/CFT

7/91 Great Expectations Prerequisites Graduate quantum mechanics including “second quantization” Ideas of “Mean Field Theory” Some familiarity with “classical” superconductivity

At the end of the lectures, you can/will have Understood concepts of VMC Know-how to write your own code A free code (pedestrianVMC) Seen examples of VMC in strongly correlated systems

8/91 References

NATO Series Lecture Notes by Nightingale, Umrigar, see http://www.phys.uri.edu/˜nigh/QMC-NATO/webpage/ abstracts/lectures.html Morningstar, hep-lat/070202 Ceperley, Chester and Kalos, PRB 17, 3081 (1977)

9/91 VMC in a Nutshell

Strategy to study ground (and some special excited) state(s) “Guess” the ground state Use Schr¨odinger’s variational principle to optimize the ground state Calculate observables of interest with the optimal state ...

10/91 Hall of Fame

Bardeen, Cooper, Schrieffer Laughlin

Quote/Question from A2Z, Plain Vanilla JPCM (2004) The philosophy of this method (VMC )is analogous to that used by BCS for superconductivity, and by Laughlin for the fractional quantum Hall effect: simply guess a wavefunction. Is there any better way to solve a non-perturbative many-body problem?

11/91 Variational Principle

Eigenstates of the Hamiltonian satisfy

H Ψ = E Ψ | i | i where E is the energy eigenvalue Alternate way of stating this ◮ Define an “energy functional” for any given state Φ | i E[Φ] = Φ H Φ h | | i ◮ Among all states Φ that satisfy Φ Φ = 1, those that extremize E[Φ] | i h | i are eigenstates of H Exercise: Prove this! ◮ The state(s) that minimises(minimise) E[Φ] is(are) the ground state(s)

12/91 Variational Principle in Action...A Simple Example

+ Ground state H2 molecule with a given nuclear separation “Guess” the (linear combination of atomic orbitals)

ψ = a 1 + b 2 | i | i | i where 1, 2 are atomic orbitals centered around protons 1 and 2 | i Find energy function (taking everything real)

2 2 E(a, b)= H11a + 2H12ab + H22b

(symbols have obvious meaning) 1 Minimize (with appropriate constraints) to find ψ g = ( 1 + 2 ) | i √2 | i | i (for H12 < 0) Question: Is this the actual ground state? Example shows key ideas (and limitations) of the variational approach!

13/91 Variational Approach in Many Body Physics

Guess the ground state wavefunction Ψ | i Ψ depends on some variational parameters α1, α2... which “contain |thei physics” of the wavefunction ◮ Example,

† † BCS = uk + vk c c 0 | i k↑ −k↓ | i " k # Y   uk , vk are the “αi s” In general, αi s are chosen ◮ To realize some sort of “order” or “broken symmetry”, (for example, magnetic order)...there could be more than one order present (for example, superconductivity and magnetism) ◮ To enforce some sort of “correlation” that we “simply” know exists and can include all of the “effects” above ...your guess is not as good as mine!

14/91 Variational Principle vs. MFT

Mean Field theory

◮ Replace H by HMF which is usually quadratic ◮ Use the Feynman identity F FMF + H HMF MF (F is free energy) ≤ h − i ◮ Kills “fluctuations” Variational Principle ◮ Works with the full Hamiltonian ◮ Non-perturbative ◮ Well designed wavefunctions can include fluctuations above MFT Usually MFT provides us with intuitive starting point...we can then “well design” the variational wavefunction to include the fluctuations that we miss

15/91 Overall Strategy of Variational Approach

Summary of Variational Approach

1 Design a ground state Ψ(αi ) | i 2 Energy function

Ψ(αi ) H Ψ(αi ) E(αi )= h | | i Ψ(αi ) Ψ(αi ) h | i m 3 Find α that minimize (extremize) E(αi ) and obtain ΨG i | i 4 Study ΨG | i

This explains the “Variational” part of VMC ... But...why Monte Carlo?

16/91 Variational Approach: The Difficulty

In many cases the variational programme can be carried out analytically where we have a well controlled analytical expression for the state, e. g., BCS state In most problems of interest we do not have a simple enough analytical expression for the wavefunction (e. g., Gutzwiller D wavefunction Ψgutz = g Ψfree ) | i | i In many of these cases we have expressions for the wavefunction which “look simple in real space”, say, Ψα(r1 , r2 ,..., rN, , r1 , r2 ,..., rN, ), where riσ are the coordinates of our↑ electrons↑ on↓ a lattice↓ ↓ (we will↓ work exclusively with lattices, most ideas can be taken to continuum in a straightforward way); Ψα has necessary permutation symmetries, and α subscripts reminds us of the underlying parameters

Call C (r1 , r2 ,..., rN, , r1 , r2 ,..., rN, ) a configuration of the ≡ ↑ ↑ ↓ ↓ ↓ ↓ electrons

17/91 Variational Approach: The Difficulty Now, Ψ (C)HΨ (C) E(α )= C α∗ α i Ψ (C)Ψ (C) P C α∗ α i.e., we need to do a sum overP all configurations C of the our 2N electron system Note that in most practical cases we need sum the denominator as well, since we use unnormalized wavefunctions! Sums...how difficult can they be? Consider 50 electrons, and 50 electrons on a lattice with 100 ↑ ↓ sites....How many configurations are there? Num. Configurations 1060!!!! ∼ And this is the difficulty! ... Solution...use Monte Carlo method to perform the sum!

18/91 “Monte Carlo” of VMC

How do we use Monte Carlo to evaluate E(αi )? Use the following trick Poor Notation Warning: HΨ(C) C H Ψ ≡ h | | i

HΨα(C) Ψ (C)HΨ (C) C Ψα∗ (C)Ψα(C) Ψ (C) E(α ) = C α∗ α = α i Ψ (C)Ψ (C) Ψ (C)Ψ (C) P C α∗ α P C α∗ α HΨ (C) = PP(C) α P Ψ (C) C α X where Ψ (C) 2 P(C)= | α | C Ψα∗ (C)Ψα(C) is the probability of the configurationP C; obviously C P(C) = 1 Note that we do not know the denominator of P(C) P For any operator O OΨ (C) O = P(C) α h i Ψ (C) C α X 19/91 Idea of Monte Carlo Sums

Key idea is not to sum over all configurations C, but to visit the “most important” configurations and add up their contribution The most important configurations are those that have “high” probability..., but we do not know the denominator of P(C)

Need to develop a scheme where we start with a configuration C1 and “step” to another configuration C2 comparing P(C1) and P(C2)...and continue on Thus we do a “walk” in the space of configurations depending on P(C)...spending “most of our time” in high probability configurations...this is importance sampling Goal: construct a set of rules (algorithm) to “step” in the configuration space that performs the necessary importance sampling Such a stepping produces a sequence configurations a function of steps or “time”...but how do we create this “dynamics”?

20/91 Markov Chains

Let us set notation: Label all the configurations (yes, all 1060) using natural numbers C , C ...Cm... etc; we will use P(1) P(C ), and in 1 2 ≡ 1 general P(m) P(Cm) and ≡ C ≡ m We construct transition (Markov) matrix W (n, m) W (n m) P P ≡ ← which defines the probability of moving from the configuration m to configuration n in a step, we consider only W s that are independent of time The probability of taking a step from m n is independent of how → the system got to the state m...such stochastic processes are called as Markov chain Clearly W (n, m) = 1, and W (n, m) 0 Exercise: If λ is a right n ≥ eigenvalue of the matrix W, show that |λ| ≤ 1. P

21/91 Master Equation Question: Given a Markov matrix W, and the knowledge that the over the configuration space at step t is P(t) (P is the column P(m)), what is P(t + 1)? Clearly P(n, t + 1) = W (n, m)P(m, t) P(t + 1) = WP(t) m X How does the probability of a state change with time? P(n, t + 1) P(n, t) = W (n, m)P(m, t) W (m, n)P(n, t) − m − m X X ...the Master equation For a “steady” probability distribution W (n, m)P(m) W (m, n)P(n) = 0 m − m X X = W (n, m)P(m) = P(n) = WP = P ⇒ m ⇒ ...i.e., P is a right eigenvectorX of the Markov matrix W with unit eigenvalue! 22/91 Markov Chains...a bit more

For our purpose, need to construct a Markov matrix W for which our P is an eigenvector...how do we do this? We define the visit probability V (n, m; t) is the probability that the system reaches state n at time t for the first time given that the system was in configuration m at time 0 Exercise: Express V(t) in terms of W

The quantity V (n, m)= t V (n, m; t) is the total visit probability of n from m P If V (n, m) = 0, n, m, i. e., any state can be reached from any other 6 ∀ state, then Markov chain is said to be irreducible

23/91 Markov Chains...a bit more

A state m is called recurrent if V (m, m) = 1, i.e., and the mean recurrence time is tm = tV (m, m; t); if tm < , the state is t ∞ called positive recurrent; if all states are positive recurrent, the chain P is called ergodic A chain is aperiodic if no state repeats in a periodic manner Fundamental theorem of Irreducible Markov Chains: An ergodic aperiodic irreducible Markov chain has a unique stationary probability distribution We now need to construct an irreducible chain, i. e., the key point is that every state must be reachable from every other state The other conditions (positive recurrence) is easier to satisfy since our configuration space is finite albeit large

24/91 Detailed Balance How do we choose an irreducible chain whose stationary distribution is our distribution? Clearly, there is no unique choice! We can choose W in such a way that

W (n, m)P(m)= W (m, n)P(n)

and this will clearly mean that P is the right eigenvector of W...this condition is called the detailed balance condition Now, we are ready to construct an appropriate W, define

W (n, m)= A(n, m)S(n, m)

where S(n, m) is called the “suggestion probability” i.e., the probability of suggesting a state n to move to, when we are at m, and A(n, m) is called the “acceptance probability” Let us suppose that we have “designed” a suggestion probability matrix...what is the acceptance probability matrix?

25/91 Acceptance Matrix

We can use detailed balance and immediately the following equation for the acceptance matrix

A(n, m) S(m, n)P(n) S(m, n) Ψ(n) 2 = = | | h(n, m) A(m, n) S(n, m)P(m) S(n, m) Ψ(m) 2 ≡ | | note that denominators are gone! We now posit A(n, m) to be function of h(n, m), i.e.,

A(n, m)= F (h(n, m))

The first condition can be satisfied if the function F satisfies

F (y)= yF (1/y)

We have lots of freedom...we can choose any function that satisfies the above condition!

26/91 Acceptance Matrix: Popular Choices

Metropolis

S(m, n) Ψ(n) 2 F (y) = min 1, y = A(n, m) = min 1, | | { } ⇒ S(n, m) Ψ(m) 2  | |  Heat Bath y S(m, n) Ψ(n) 2 F (y)= = A(n, m)= | | 1+ y ⇒ S(m, n) Ψ(n) 2 + S(n, m) Ψ(m) 2 | | | |

27/91 Suggestion Matrix Simplest suggestion matrix: Based on single electron moves Single electron moves: Successive configurations differ only in the position of a single electron ◮ At any step pick an electron at random ◮ Move the electron from its current position to a new position also picked at random ◮ We have 1 S(n, m) = Nelectrons (Nsites 1) × − Note that S(m, n) = S(n, m) Metropolis acceptance matrix becomes Ψ(n) 2 min 1, | | Ψ(m) 2  | |  ...we have constructed our Markov matrix W Question: Does our W produces an irreducible chain?

28/91 ..and some Final Touches

We now generate a sequence of configurations mk starting from some random initial configuration using our Markov matrix W...say we take NMC steps The key result of importance sampling

NMC 1 HΨα(mk ) E(αi )= NMC Ψ (mk ) k α X=1 ...and this work for other operators as well

NMC 1 OΨα(mk ) O α = h i NMC Ψ (mk ) k α X=1

29/91 Variational Monte Carlo

VMC Summary 1 Start with a random configuration 2 For k = 0, NMC ◮ In configuration mk , pick an electron at random, suggest a random ′ new position for this electron, call this configuration mk ′ 2 ◮ ′ |Ψ(mk )| ′ Accept m as mk with probability min 1, 2 , if m is not k +1 |Ψ(mk )| k accepted, set mk+1 = mk n o ◮ Accumulate OΨα(mk+1) Ψα(mk+1) 3 Calculate necessary averages for E(αi )

4 Optimize over αi , obtain ΨG | i 5 Study ΨG | i

30/91 Comments on VMC

Note that our starting configuration in most practical situations is sampled from a different probability distribution from the one we are interested in! If we take sufficiently large number of steps we are guaranteed by the fundamental theorem that we will “reach” our desired distribution How fast we attain our desired distribution depends next largest eigenvalue of W...it might therefore be desirable to design W such that all other eigenvalues of W are much less than unity Passing remark: In VMC we know which P we want and we construct a W to sample P...In “Green’s Function Monte Carlo” (GFMC) we have a W = e H , we start with some P( Ψ ) and project out the − ≡| i ground state PGroundState ( ΨG ) by repeated application of W ≡| i

31/91 VMC Computational Hardness

How hard is VMC? For every step, once we make a suggestion for a new configuration, ′ 2 Ψ(mk ) we need to evaluate the ratio | |2 Ψ(mk ) | | Now Ψ is usually obtained in a determinental form. If there are 2N electrons, we usually have determinants of order N N × How hard is the calculation of a determinant? This is an N3 process... Not too good, particularly since we have to optimize with respect to αi ...we like things to be O(N) hard! ...... all is not lost!

32/91 Faster VMC Note that throughout our calculation, we need only the ratio of determinants, not the determinants themselves To illustrate the idea behind this, consider spinless fermions; we have a configuration m given by a wavefunction Ψ and a configuration n given by a wavefunction Φ which differ by| thei position of a single | i electron l, rl and rl′; written in terms of matrices

ψa1 (r1) ψa1 (rl ) ψa1 (rN ) . · · · . · ·. · [Ψ] =  . . .  · · · ψa (r ) ψa (rl ) ψa (rN )  N 1 · · · N · · · N    and

′ ψa1 (r1) ψa1 (rl ) ψa1 (rN ) . · · · . · ·. · [Φ] =  . . .  · · · ′ ψa (r ) ψa (r ) ψa (rN )  N 1 · · · N l · · · N    where ψai are some one particle states (which usually depend on αi )

33/91 Faster VMC det[Φ] We need det[Ψ] ...key point:[Φ] and [Ψ] differ only in one column! Now

−1 Φ Φkj = δij = CofΦki Φkj = det[Φ]δij ik ⇒ k k X X = CofΦkl Φkl = det[Φ] ⇒ k X CofΦkl = CofΨkl = det[Φ] = CofΨkl Φkl ⇒ k X det[Φ] −1 = = Ψ Φkl ⇒ det[Ψ] kl k X which is certainly O(N) 1 Note that we need to know Ψ− , and thus have to store it at every step! When we accept a suggested configuration, we will have to 1 3 recalculate Ψ− which, again, appears to be O(N )!

34/91 1 Fast Update of Ψ− (Sherman-Morrison-Woodbury) On acceptance of the suggested configuration, we need to compute 1 1 Φ− which will be the new Ψ− for the next step det[Φ] Calling r = det[Ψ] , we immediately have 1 Φ 1 = Ψ 1 lj− r lj− 1 Other rows of Φ− can be obtained as

1 1 1 1 1 Φij− =Ψij− βi Ψlj− , βi = Ψik− Φkl i=l − r 6 k X which is certainly an O(N2) process Note that care has to be exercised in using this procedure to update 1 Ψ− due to accumulation of (round-off) errors...every so often we 1 must generate Ψ− from the positions of the electrons using an O(N3) algorithm In short VMC, at its best, is O(N2) hard! Note that calculation of correlation functions can be harder 35/91 Variational Monte Carlo

VMC Summary 1 Start with a random configuration 2 For k = 0, NMC ◮ In configuration mk , pick an electron at random, suggest a random ′ new position for this electron, call this configuration mk ′ 2 ◮ ′ |Ψ(mk )| ′ Accept m as mk with probability min 1, 2 , if m is not k +1 |Ψ(mk )| k accepted, set mk+1 = mk n o ◮ −1 ′ Update Ψ if mk+1 = mk using SMH formula (check every so often and perform a full update) ◮ Accumulate OΨα(mk+1) Ψα(mk+1) 3 Calculate necessary averages for E(αi )

4 Optimize over αi , obtain ΨG | i 5 Study ΨG | i

36/91 Free VMC Code pedestrianVMC

37/91 pedestrianVMC Examples We will work on the square lattice, us- ing a tilted configuration, number of sites go as L2 +1 (L has to be odd)

(Paramekanti, Randeria and Trivedi, PRB (2004)) Hubbard model

tij ci† cjσ + U ni ni − σ ↑ ↓ ij i Xh i X we will consider two cases: U = 0 (free electrons) and U = ∞ Number of electrons in the lattice is characterized by x the doping from the half filled state (one electron for each lattice site

38/91 Free Gas (U = 0) Guess for the ground state Ψ = FS | i | i Suggestion matrix using single electon moves

-1.45

MC Exact -1.5

-1.55 Energy per site

-1.6

0 0.1 0.2 0.3 0.4 x (hole doping)

L = 9, NMC = 4000 55 ...note we have sampled only 10− of the possible configurations! 39/91 Extremely Correlated Liquid (U = ) ∞ Guess Ψ = P FS – Projected Fermi Liquid...In terms of real space | i | i wavefunctions the P means this: configurations C with at least one ∀ doubly occupied site the wave function Ψ(C) vanishes...double occupancy is strictly forbidden Suggestion matrix using single electon moves ...does NOT work! Need to introduce spin flip moves ... this is actually a two-electron move 0

-0.5

Projected -1 Free Energy per site

-1.5

0 0.1 0.2 0.3 0.4 x (hole doping)

L = 9, NMC = 16000

40/91 How well are we doing?

Sampling Error Systematic Error (Random number generator etc.) Finite Size Effects ... Bugs

41/91 How well are we doing? We use the estimator

1 OΨ(mk ) O = Ok , Ok h i NMC ≡ Ψ(mk ) k X for the expectation value of the operator 0.6

0.5

0.4

0.3

0.2 Energy per site

0.1

0

-0.1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Monte Carlo Sweeps We are sampling a finite chain... leads to sampling error

42/91 Sampling Error

Suppose we have a Gaussian distribution, want to find its mean value by sampling...if we perform N independent samplings, we can estimate the mean with a standard error of σ/√N where σ is the variance of the Gaussian In our case we can estimate the error as

2 2 2 1 1 εO = Ok Ok NMC − NMC * k ! + * k + X X 1 = (Oj O )(Ok O ) N2 h − h i − h i i MC jk X 1 2 1 = (Ok O ) + 2 (Oj O )(Ok O ) NMC h − h i i N h − h i − h i i k ! MC j k X X6= NMC 1 O 2 1 k O = (σ ) + (1 )λk NMC NMC − NMC k X=1

43/91 Sampling Error Autocorrelation function O λ = (Ok O )(O O ) k h − h i 0 − h i i O O clearly, σ = λ0 We see in additionq to the variance, the auto correlation function also contributes to the errors Always look at auto-correlation functions

1

0.75

0 0.5 λ / k λ

0.25

0

0 5 10 15 20 k (MC sweep)

U = ∞, x = 0.1, L = 9 Not bad! 44/91 Sampling Error: Blocking

There is a nice trick to calculate the error of auto-correlated data without calculating autocorrelations...this is called blocking...attributed to Wilson The idea is same as Kadanoff’ block spin RG trick... Start with the data of size NMC which for convenience we can take to be a power of 2, calculate its standard error ε0

Now generate new data which is of size NMC /2 by blocked Ok = (O2k + O2k+1)/2 and calculate its standard error ε1...the subscript “1” stands for first level of blocking

Continue this procedure (you need sufficient data) and plot εb as a function b where b is the number of block transformations applied Pick the plateau value for the error

45/91 Sampling Error: Blocking Our example, we are doing okay

4.0x10-04

3.5x10-04

3.0x10-04

2.5x10-04

2.0x10-04 Standard Error 1.5x10-04

1.0x10-04

0 2 4 6 8 10 Number of blocks

15 U = ∞, x = 0.1, L = 9, NMC = 2 An example with strong auto-correlations

(Flyvbjerg and Petersen, JCP (1984))

46/91 Equilibration

How to check for equilibration...run calculations with different random seeds!

-0.253 Colours stand for different random seeds

-0.254

-0.255

Energy per-0.256 site

-0.257

103 104 105 nmc (Number of Monte Carlo Sweeps)

U = ∞, x = 0.1, L = 9 “Large” seed-dependence is a sure sign of non-equilibration...we are doing okay when NMC 10000 in this example ≥

47/91 One Final Check How well can we do? If everything is okay the error should fall of as 1/√NMC

1.0x10-03

5.0x10-04 ε

103 104 105

Nmc

U = ∞, x = 0.1, L = 9 We are doing pretty okay!...But, are we really?

48/91 Another Difficulty

When we want to optimize with respect to the variational parameters, we will have to evaluate the energy many times over...long that reduce error bars are prohibitive Shorter runs produce data that are bumpy, and seed dependent...

The problem is particularly severe if the minimum is “shallow”...and becomes worse with increasing number of variational parameters

49/91 Difficulty Overcome: Parallelization

“Seed parallelization” – Cloning -0.8605 × 4 4 NMC = 5 10 Neq = 10 Quadratic Fit µ = -0.7 m = 0 N 2 Nsites = 9 + 1 x = 0.32 -0.8610

-0.8615 Energy per site

-0.8620 0.3 0.4 0.5 0.6 ∆

Produces “smooth” enough data (we use up to 120 clones)

50/91 Optimizing Variational Parameters

No derivative information ◮ Direct search ◮ Simplex method (use amoeba routine from Numerical Recipes) ◮ Genetic algorithms? Method with derivative information

51/91 Results of Force Method Seed average forces -0.8605 N = 5×104 N = 104 MC 4 eq Quadratic Fit Neq = 10 µ = -0.7 µ = -0.7 m = 0 N 2 0.0050 m = 0 N = 9 + 1 N 2 sites N = 9 + 1 x = 0.32 sites -0.8610 x = 0.32

∆ 0.0000 dE/d -0.8615

Energy per site -0.0050

× 4 NMC = 5 10 Forces from quadratic fit to energy Zero reference -0.8620 -0.0100 0.3 0.4 0.5 0.6 0.3 0.4 0.5 0.6 ∆ ∆ Results for optimized variational parameters

52/91 Variational Monte Carlo VMC Summary 1 Start with a random configuration 2 For k = 0, NMC ◮ In configuration mk , pick an electron at random, suggest a random ′ new position for this electron, call this configuration mk ′ 2 ◮ ′ |Ψ(mk )| ′ Accept m as mk with probability min 1, 2 , if m is not k +1 |Ψ(mk )| k accepted, set mk+1 = mk n o ◮ −1 ′ Update Ψ if mk+1 = mk using SMH formula (check every so often and perform a full update) ◮ Accumulate OΨα(mk+1) Ψα(mk+1) 3 Calculate necessary averages for E(αi )

4 Optimize over αi , obtain ΨG | i 5 Study ΨG | i

Finally, our answers will be only as good as our guess...

53/91 Applications to Superconductivity: Overview

Essential High Tc Models etc.. RVB Wavefunction + VMC Results Material dependencies of the phase diagram Graphene

54/91 References on VMC in High Tc

Gross et al., Giamarchi et al., Ogata et al., TKLee et al., PLee et al. and others Paramekanti, Randeria, Trivedi, PRB (2004) (see also, earlier PRL) (A2Z) Anderson, Lee, Randeria, Rice, Trivedi, Zhang, JPCM (2004) Recent comprehensive review: Edegger, Muthukumar, Gros Adv. Phys (2007)

55/91 Cuprate Phase Diagram Generic phase diagram

(Damascelli et al. 2003) Undoped system is a Mott insulator Note the electron-hole asymmetry

56/91 Cuprates: Phase Diagram

Key features T

“Strange” Metal

Pseudogap Phase max Tc ? d-Superconductor Antiferromagnet

xAF xo x

xAF – doping at which AF dies max Tc – maximum superconducting Tc xo – optimal doping

57/91 Model Hamiltonian What is the minimal model? One band model (Anderson et al.) Undoped system is a Mott insulator...must have large onsite repulsion Minimal model - one band Hubbard model

HH = tij (ci† cjσ + h.c.) µ niσ +U ni ni − σ − ↑ ↓ ij iσ i Xh i X X K Related tJ model| {z }

HtJ = P tij (c† cj + h.c.)P µ ni − iσ σ − σ ij iσ Xh i X 1 + J Si Sj ni nj · − 4 ij   Xh i

HJ P stands for projection| – hopping{z should not} produce doublons 58/91 Are the two Models Related? Apply a canonical transformation on the Hubbard model to “project out” doubly occupied sector of the Hilbert space iS iS Canonical transformation e He− ...this leads to

PHH P = PKP + HJ + HK3site O(t2/U) J is now related to t and U as | {z } 4t2 J = U can be understood from a two site model Typically U 10 12t, J/t 0.3, t 0.3 0.5eV ∼ − ∼ ∼ − Note that U gives us → ∞ HU= = PKP ∞ which we had seen earlier 59/91 U = Revisited ∞

Nothing can hop at half filling...what happens at a doping x? Ask the question: What is the relative probability an electron will find 2x a hole on the neighbouring site?...this can be shown to be gt = 1+x Suggests that

H = PKP gt K! 7→ which is now a “free particle” Hamiltonian! How well does this do?

60/91 U = Revisited ∞ Our guess for the ground state is Ψ = P FS ... | i | i The “projected kinetic energy” is Ψ H Ψ = gt FS K FS h | | i h | | i 0

-0.5

Projected -1 Free

Energy per site × gt Free

-1.5

0 0.1 0.2 0.3 0.4 x (hole doping)

L = 9 Does okay?...looks like!

61/91 U = and the tJ model ∞

From the U = results, we see immediately that kinetic energy is ∞ badly “frustrated”

What happens when we add the HJ to PKP?...this is the tJ model At zero doping (x = 0) we get

HtJ = J Si Sj · x=0 ij Xh i which is the Heisenberg model...|{z} On a square lattice we know this has a Ne´el ground state (with a smaller staggered magnetization)

62/91 tJ model...effect of doping

Projection badly frustrates kinetic energy...what is the effect of HJ on kinetic energy?

Hj wants singlets...but this is “localizing” and frustrates the kinetic energy further... Expect ◮ J to win over t at “small” doping...very little kinetic energy to be gained...lots of J energy to gained ◮ At “large” doping expect kinetic energy to win ◮ Doping where competition is most severe 2xt J = x 0.15!! ∼ ⇒ ∼ Who wins?

63/91 Ground State at Finite Doping

Actually, both, in a superconducting state!

Both J and t can be satisfied by forming resonating singlets Idea embodied in the Resonating Valance Bond (RVB) state, Anderson (1987) How do we construct an RVB state? Will (can) it be a superconductor of the right pairing symmetry?

64/91 Mean Field Theory Keep projection aside...rewrite the tJ Hamiltonian as

HtJ = K J b† bij − ij ij X where b† (c† c† c† c† ) ij ∼ i j − i j Construct a meanfield↑ ↓ Hamiltonian↓ ↑

HMF = K ∆ij bij + ... − ij X

where ∆ij J b† ... is the “singlet pair amplitude” ∼ h ij i Translational invariance two pair amplitudes ∆x and ∆y — d symmetry is obtained by keeping ∆x = ∆ and ∆y = ∆ − We now have a BCS like Hamiltonian, ground state

dBCS = (uk + vk c† c† ) 0 | i k k | i k ↑ − ↓ Y 65/91 dRVB state

The “guess” for the ground state is

† † P φk c c vk Ψ = P dBCS = e k k↑ −k↓ 0 φk = | i | i | i uk For working on a lattice

N/2

Ψ = P φk c† c† 0 | i k k | i k ↑ − ↓! X N/2 = P φ(i j)c† c† 0 − i j | i k ↑ ↓! X All our variational parameters are buried in φ(i j), the “pair − wavefunction”

66/91 Variational Wavefunction

Given variational parameters construct φ N Pick number of electrons N using doping x = 1 (NL number of − NL sites) Variational wave function

φ(r1 r1 ) ... φ(r1 rN/2 ) ↑ .− ↓ . ↑ −. ↓ PΨ( ri , rj )= . .. . { ↑ ↓} φ(rN/2 r1 ) ... φ(rN/2 rN/2 ) ↑ − ↓ ↑ − ↓

Optimize, obtain the ground state, and study it

67/91 Superconducting order parameter

The superconducting correlation function

F (r r′)= br† br′ α,β − h α βi 1 br† = (c† c† c† c† ) creates a singlet on the bond (r, r +α ˆ), α 2 r r+ˆα − r r+ˆα α = x, y ↑ ↓ ↓ ↑ SC order parameter

Φ = lim Fα,α(r r′) r r′ − | − |−→∞ Off-diagonal Long Range Order (ODLRO)

68/91 Results from Paramekanti et al.

Calculation includes three site hopping

Why does Φ go to zero at x?...key result of projection!

69/91 More results from Paramekanti et al.

Seems to agree qualitatively with many experiments What about antiferromagnetism?

70/91 Cuprates: Our Focus

Is there a transition from antiferromagnetic ground state to a superconducting state? What is the nature of this transition? Do antiferromagnetism and superconductivity co-exist? (Previous work: Gros et al, 1988, Giamarchi et al 1990, Ogata et al, 1995, Lee et al. 1995-2003) What is the origin of the electron-hole asymmetry? Why does electron doping ”protect” the antiferromagnetic state? What energy scales determine the answers to the above questions? Does the next-near neighbour and third-near neighbour hopping affect the ground state and the transition? How? There is a suggestion (Pavarini et al., PRL, 2001)) based on the results of max first principles calculations that Tc is correlated with the ratio of the second neighbour hopping to that of the nearest neighbour hopping? What is the microscopic reason behind this? Is there a “single parameter” description of the phase diagram that captures the “material dependencies”?

71/91 T max Material Dependence of c

Range parameter r = t′/t, t′′ = t′/2 (Pavarini et al., 2001)

max Correlation between Tc and r, r obtained from ab initio electronic structure methods

72/91 Meanfield Theory

† 1 † 1 S S c σ c c σ ′ ′ c ′ Meanfield breakup up of i · j = “ iα 2 αβ iβ ” · “ jα′ 2 α β jβ ” Neel channel

Si · Sj ∼ hSi iSj + ...

iRi ·Q with hSi i = mN ez e , Q = (π, π) d-wave channel

S S c† c† c† c† ... i · j ∼ ∆ij “ i↑ j↓ − i↓ j↑” +

with ∆ij = ∆ on x-bonds and ∆ij = −∆ on y-bonds “Fock”-channel

† Si · Sj ∼ χij ciσcjσ + ...

′ ′′ with χij = χ , χ ... “renormalizes” kinetic energy ′ ′′ Four “molecular field parameters”: mN , ∆, χ , χ

73/91 Meanfield Theory

Mean field Hamiltonian

† H = ξ(k)c ck MF X kσ σ k,σ 1 + ∆(k) c† c† + h. c. X “ k↑ −k↓ ” 2 k

1 † + σmN J(Q)c ckσ 2 X k+Q,σ kσ

contains five parameters χ , χ , ∆, mN , µ – gives the “one particle” { ′ ′′ } states ∆: Gap parameter

mN : Neel order parameter µ: Hartree shift

χ′, χ′′ : Fock shifts

74/91 Variational Ground State Construction

Constructed in two steps ◮ Diagonalize the Hamiltonian without the paring term...give rises to two † SDW bands dkα which disperse as Eα(k) ◮ Introduce pairing in the two spin split bands...no “cross pairing” terms arise Construct a BCS wavefunction

vkα ∆(k) |Ψi = (u + v d† d† )|0i, = (−1)α−1 , α = 1, 2 Y kα kα kα↑ −kα↓ uk k 2 k 2 k kα α Eα( )+ qEα( ) + ∆ ( )

Project to N particle subspace

N/2 |Ψi = 0 ϕ(R , R )c† c† 1 |0i N X i↑ j↓ i↑ j↓ @ i,j A

The “pairing function” ϕ(Ri , Rj ) is determined by the parameters mN , ∆...

75/91 Electron Doping

Electron doped Hamiltonian is obtained by a particle-hole transformation (Lee et al. 1997, 2003, 2004) c c on A sublattice, c c on B sublattice † −→ † −→ − Corresponds to t t, t t , t t −→ ′ −→ − ′ ′′ −→ − ′′ Choose sign convention such that holed doped side has all ts positive...electron doped side will have t′ and t′′ negative...

76/91 What do we measure?

Staggered magnetization 2 M = Sz Sz N h i − i i i A i B Xǫ Xǫ The superconducting correlation function (Paramekanti et al. 2004)

F (r r′)= Br† Br′ α,β − h α βi 1 Br† = (c† c† c† c† ) creates a singlet on the bond (r, r +α ˆ), α 2 r r+ˆα − r r+ˆα α = x, y ↑ ↓ ↓ ↑ SC order parameter

Φ = lim Fα,α(r r′) r r′ − | − |−→∞ Off-diagonal Long Range Order (ODLRO)

77/91 Phase Diagram (t′′ = 0)

100 100   M , t’ = 0.0 Φ t’’/t = 0.0, J/t = 0.3 , t’ = 0.1 ) t’ = 0.2 )

-4 , 80 0.8 80 -4 , t’ = 0.3 10 10 × × ( 60 60 ( Φ Φ M 40 t’ < 0 0.4 t’ > 0 40

20 20 ODLRO, ODLRO,

0 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Electron Doping, x Hole Doping, x

SC and AF coexist on both h and e doped systems With increasing t , SC enhanced on h doped side, AF on the e | ′| doped side ((Lee et al, 2004, Trambley et al, 2006, Kotliar et al 2007))

New result: t does not affect xAF on the hole doped side, and | ′| slightly “weakens” SC on the e doped side

78/91 Phase Diagram (t′′ = 0) 6 100 100   M t’/t = -0.3, J/t = 0.3 , t’’ = 0.00 Φ t’/t = 0.3 , t’’ = 0.03 ) )

80 0.8 , t’’ = 0.06 80 -4 -4 , t’’ = 0.12 10 10 t’’ = 0.15 ×

, × ( 60 0.6 60 ( Φ Φ M 40 t’’ < 0 0.4 t’’ > 0 40

20 0.2 20 ODLRO, ODLRO,

0 0 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Electron Doping, x Hole Doping, x

h-doped: Increasing t enhances SC, xAF is unaffected | ′′| e-doped: Increasing t enhances AF, SC is unaffected | ′′| ... Is there a “single parameter” characterization...a unified picture?

79/91 Fermi Surface Convexity Parameter η 1 1

0.5 0.5

kF π π

θ / /

y 0 y 0 k k

-0.5 -0.5

-0.5 0 0.5 1 -0.5 0 0.5 1 π k /π kx/ x Bare Fermi surface (BFS) at zero doping Fermi surface convexity parameter

k NODE 2 η = 2 F kANTINODE  F 

η = 1 when t′ = t′′ = 0 η > 1 convex BFS, η < 1 concave BFS Claim: “Everything” is determined by η (for given t and J); Convex BFS encourages AF, concave BFS promotes SC!

80/91 Fermi Surface Convexity Parameter η

0.50 η 4.0 t’’= t’/2 t’’ = t’/2 3.6 3.0 3.0

η 2.0 0.00 2.4 t’’/t 1.8 1.0 1.2 -0.50 -0.50 0.00 0.50 0.6 -0.50 0.00 0.50 t’/t t’/t

η > 1 convex BFS, η < 1 concave BFS

For t′′ = t′/2: η is monotonically related to range parameter of Pavarini et al.! Valid for t = t /2 ′′ 6 ′

81/91 “Extent” of antiferromagnetism xAF and η

0.30 Electron Doping

AF 0.25 Hole Doping

0.20 t’= 0.0, t’’=0.00 0.15 t’=-0.3, t’’=0.15

t’=0.3, t’’=0.06

AF extent, x 0.10 t’=0.4, t’’=0.00 0.5 1.0 1.5 2.0 2.5 η

Convex BFS promote AF, xAF increases linearly with η!

Different values of t′, t′′ with same η fall on top of each other!

82/91 Optimal ODLRO Φmax and η

50 HgBa Ca Cu O Electron Doping 140 2 2 3 8 HgBa2CaCu2O6 ) Tl Ba Ca Cu O 40 Hole Doping 120 2 2 2 3 10 -4 100 10 HgBa CuO 30 2 4 YBa2Cu3O7 × 80 Tl2Ba2CuO6 ( c max

20 T La CaCu O 60 2 2 6 LaBa2Cu3O7 max TlBaLaCuO d 5 La CuO 40 2 4 Φ 10 Bi2Sr2CuO6 Pb Sr Cu O 20 2 2 2 6 Ca2CuO2Cl2 0.5 1.0 1.5 2.0 2.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 η η

The maximum ODLRO (Φmax ) is promoted by a concave BFS

Φmax taken to be a measure of Tc , suggests η “determines” Tc ! Consistent with experiments, Pavarini et al., 2001

83/91 Physics of η

Key (well known) point: t′ and t′′ do not disturb the AF background

Question: Do t′, t′′ help gain kinetic energy?

84/91 Physics of η Not always! Look at the bare kinetic energy as fn. of doping -1.0 η = 3.0 η = 2.0 η = 1.0 -1.2 η = 0.7  η = 0.6 Concave FS  -1.4

-1.6  Convex FS  -1.8 Bare kinetic energy per site

-2.0 0.0 0.1 0.2 0.3 0.4 Doping x m xm

For η > 1, the bare KE falls with increasing doping, attains a BKE minimum at xm BKE xm increases with η This gain in KE arises from t′, t′′ and hence AF will be stable until BKE xm ...explains why xAF increases with η! 85/91 Physics of η

-1.0 η = 3.0 η = 2.0 η = 1.0 -1.2 η = 0.7  η = 0.6 Concave FS  -1.4

-1.6  Convex FS  -1.8 Bare kinetic energy per site

-2.0 0.0 0.1 0.2 0.3 0.4 Doping x m xm

For η < 1, the bare KE monotonically increases with doping...greater rate of increase of BKE for more concave FS... Plausible that SC is stabilized in this case...both KE and exchange energies are happy!

86/91 Summary What is done VMC of t t t J model...a detailed study of material − ′ − ′′ − dependencies An efficient method for optimizing variational wavefunctions

What is learnt Cuprate phase diagram is characterized by a single parameter...η (for given t and J) Difference between hole and electron doping Microscopic origin of Pavarini et al. relation

Suggestion to increase Tc : Make BFS more concave!

Reference S. Pathak, V. B. Shenoy, M. Randeria, N. Trivedi, Phys. Rev. Lett. 102, 027002 (2009)

87/91 Graphene: Superconductor?

Work done in collaboration with G. Baskaran Graphene - many benzene’s connected togather Pauling’s idea of resonance Graphene: Correlated or not? Experiments suggest a U/t 2.4 ∼ Not strongly correlated

88/91 Graphene Superconductivity

Singlet formation tendency...J (difference between singlet triplet energies) Undoped graphene has zero density of states...MFT suggests that a critical J is required to induce SC J in graphene is lower than the critical value, undoped graphene is not a SC What about doped graphene? Meanfield theory of Black-Schaffer suggested a d + id state

89/91 Graphene Superconductivity: VMC Results

D Ψ = g BCSd+id

| ) i6 | i 25 -4 10 × ( 20 Φ 4 15

10 2

5 SC Correlation function, F(r) SC Order0 Parameter, 0 0 0.1 0.2 0.3 2 3 4 5 6 7 Doping, x Distance, r Optimum doping around x 0.15 ∼ Agrees with other works e.g., Honerkamp, PRL (2008)

90/91